A modi ed fth-order WENO scheme for hyperbolic ... · PDF fileA modi ed fth-order WENO scheme for hyperbolic conservation laws Samala ... at the critical points where the rst derivatives

A modified fifth-order WENO scheme forhyperbolic conservation laws

Samala Rathan∗, G Naga Raju †

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, India

Abstract

This paper deals with a new fifth-order weighted essentially non-oscillatory (WENO)scheme improving the WENO-NS and WENO-P methods which are introduced in Ha etal. J. Comput. Phys. (2013) and Kim et al., J. Sci. Comput. (2016) respectively. Thesetwo schemes provide the fifth-order accuracy at the critical points where the first derivativesvanish but the second derivatives are non-zero. In this paper, we have presented a schemeby defining a new global-smoothness indicator which shows an improved behavior over thesolution to the WENO-NS and WENO-P schemes and the proposed scheme attains optimalapproximation order, even at the critical points where the first and second derivatives vanishbut the third derivatives are non-zero.

Keywords: Hyperbolic conservation laws, WENO scheme, smoothness indicators, non-linearweights, discontinuity.MSC: 65M20, 65N06, 41A10.

1 Introduction

The hyperbolic conservation laws may develop discontinuities in its solution even if the initialconditions are smooth. Therefore classical numerical methods which depend on Taylor’s expan-sion fails, so the spurious oscillations occur in the solution. To resolve this, the Total VariationDiminishing (TVD) schemes are constructed by Harten in [10, 11] based on the principle that, thetotal variation to the approximation of numerical solution must be non-increasing in time. Butthe TVD schemes are at most first order accurate near smooth extrema [20].

In order to overcome this difficulty, Harten et al. [12, 13, 14] succeeded by relaxing the TVDcondition and allowing the spurious oscillations to occur in the numerical scheme, but the O(1)Gibbs-like phenomena are essentially prevented. This is the first successful higher order spatial dis-cretization method for the hyperbolic conservation laws that achieves the essentially non-oscillatory(ENO) property and known as ENO schemes. In [12], finite-volume ENO method was studied andshown that, to have a uniform high-order accuracy right up to the location of any discontinuity.Later, the finite-difference ENO scheme was developed by Shu and Osher in [24, 25].

The ENO method works based on the idea of choosing the interpolation points over a stencilwhich avoids the initiation of oscillations in the numerical solution. To do this, a smoothness

∗Email: [email protected]†Email: [email protected]

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indicator of the solution is determined over each stencil and by using this, the smoothest candidatestencil is chosen from a set of candidate stencils. As the result, the ENO scheme avoids spuriousoscillations near discontinuities and obtains information from smooth regions only.

The weighted ENO (WENO) scheme is introduced by Liu et al. [19], in the finite-volumeframework up to third-order of accuracy and later, Jiang and Shu [17], introduced this WENOscheme in finite-difference framework which we refer here as WENO-JS scheme by constructing thenew smoothness indicators that measure the sum of the normalized squares of the scaled L2 normsof all derivatives of local interpolating polynomials. Very high order schemes are constructed in[2] based on the WENO-JS scheme which satisfies monotonicity preserving property.

The main idea behind the WENO scheme is that it uses a convex combination of all the ENOcandidate sub-stencils in a non-linear manner and assigns a weight to each sub-stencil between 0and 1 based on its local smoothness indicator. The basic strategy of assigning the weights is that,by the combining of the lower order polynomials with optimal weights, which yields an upwindscheme of maximum possible order in smooth regions of the solution and assigns smaller weightsto those lower order polynomials whose stencils contain discontinuities so that the essentially non-oscillatory property is achieved. The detailed description of the methodology about ENO andWENO schemes and their implementation can be found in [22, 23].

It is first pointed out by Henrick et al. in [15], that the desired convergence rate of the fifth-order WENO-JS was not achieved for many problems where the first and third derivatives of theflux do not simultaneously vanish and the behavior of the scheme is sensitive to ε, a parameteris introduced in order to avoid division by zero in the evaluation of non-linear weights. Theysuggested an improved version of the WENO-JS scheme which is termed as mapped WENOand abbreviated by WENO-M. By using a mapping function on the nonlinear weights, WENO-Msatisfies the sufficient condition where WENO-JS fails and obtains an optimal order of convergencenear simple smooth extrema. Consequently, higher order WENO schemes were developed basedon mapping function in [7].

In a different approach to the construction of the nonlinear weights, Borges et al. [3], introducedthe fifth-order WENO-Z scheme. In their study, the authors measured the smoothness of the largestencil which comprises all sub-stencils and incorporated this in devising the smoothness indicatorsand nonlinear weights. The resulting WENO-Z scheme is less dissipative than WENO-JS andWENO-M. The convergence order of the WENO-Z scheme is four at the first-order critical pointsand degrade to two when higher order critical points are encountered. Further, a closed-formformula was derived in [4] for the WENO-Z scheme to the all odd orders, higher than fifth-orderaccuracy.

The smoothness indicators based on Lagrange interpolation polynomials are derived in [5]which gives the desired order of convergence if the first and second derivatives vanish but the thirdderivatives are non-zero by constructing the higher-order global smoothness indicator in L2−sense.The resulting scheme shows less dissipative nature than other schemes and subsequently high orderschemes were presented in [6]. Modified smoothness indicators of WENO-JS scheme is presentedin [16] based on the linear combination of second order derivatives over the global stencil byusing the idea of WENO-Z scheme. Acker et al. in [1], have observed that the improving theweights information where the solution is non-smooth is more important than the improving theaccuracy at critical points. To do this, they introduced an extra term to the WENO-Z weightsso that the scheme behaves with the same stability and sharpness as the WENO-Z scheme atdiscontinuities and shocks with a higher numerical resolution. The WENO methodology is stillin development to improve its rate of convergence in smooth regions and decrease the dissipationnear the discontinuities even it is successful in a wide number of applications.

2

Recently Ha et al. [9] introduced the local as well as global smoothness indicators based onL1-norm approach and termed this scheme as WENO-NS. The WENO-NS scheme provides animproved behavior compared to other fifth-order WENO schemes for the problems which containdiscontinuities. It is well known that the smoothness indicators constructed based on L1−normapproach may lead to provide loss of regularity of the solution. To overcome this difficulty, theauthors constructed local smoothness indicators by developing an approximation method to deriva-tives with higher accuracy and introduce a parameter ξ in calculating the local smoothness indi-cators to the tradeoff between the accuracy around the smooth and discontinuous regions. Theglobal smoothness indicator is constructed as an average of the smoothness of a global stencil dataand a middle stencil information through a mapping function which satisfies desirable sufficientcondition so that the scheme achieves required order of accuracy.

The main difficulty in WENO-NS scheme is to find global smoothness indicator and because ofthe symmetry nature of the two local smoothness indicators in the WENO-NS scheme out of three,the given three sub-stencils may provide the unbalanced contribution to the evaluation of flux atthe interface. Kim et al. [18], made a balanced tradeoff among all the sub-stencils by introducing aparameter δ in the formulation of local smoothness indicators and constructed a global smoothnessindicator which does not include an extra information like as in WENO-NS scheme i.e., middlestencil data. This modification yields better performance than the WENO-NS scheme and termedas WENO-P scheme.

A simple analysis verifies that the resulting WENO-NS and WENO-P schemes have the fifth-order accuracy if the first-order critical point vanish but the second derivatives are non-zero. Themain aim of this study is to further improve the WENO-NS and WENO-P schemes to achievedesired order of accuracy even if the first and second derivatives vanish but the third derivative isnon-zero.

In this article, we analyzed the fifth-order WENO scheme with the smoothness indicatorsdeveloped in [17, 15, 3, 9, 18] and derived a new global smoothness indicator which is a linearcombination of second order derivatives leads to give a fourth-order of accuracy so, the resultingglobal smoothness indicator provides a much smoother information to the evaluation point inevaluating the non-linear weights. It is verified that the proposed WENO scheme has the fifth-order accuracy even at critical points where first and second derivatives vanish but the thirdderivative is non-zero through the Taylor expansion. We call this scheme as modified WENO-P(MWENO-P) which takes almost the same computational cost as that of WENO-NS or WENO-Pand is simple to implement as the WENO-JS or WENO-Z schemes. The numerical experimentsare shown that the proposed scheme MWENO-P performs better than WENO-JS, WENO-Z,WENO-NS and WENO-P for the problems which contain discontinuities.

The organization of the paper is as follows. Preliminaries to understand about WENO recon-structions to the one-dimensional scalar conservation laws are presented in section 2 and in section3 details about the construction of a new global measurement which estimates smoothness of alocal solution in the construction of a fifth-order WENO scheme. Some numerical results are pro-vided in Section 4 to demonstrate advantages of the proposed WENO scheme. Finally, concludingremarks are given in Section 5.

2 WENO schemes

Consider the general form of conservation law

ut + f(u)x = 0,−∞ < x <∞, t > 0, (1)

3

with initial conditionu(x, 0) = u0(x).

Here the function u = (u1, u2, ......, um)T is a m-dimensional vector of conserved variables, flux f(u)is a vector-valued function of m components, x and t denote space and time variables respectively.The system is called hyperbolic if all the eigen values λ1, λ2, ..., λm of the Jacobian matrix A = ∂f

∂u,

of the flux function are real and set of right eigen vectors are complete.Let {Ij} be a partition of a spatial domain with the jth cell Ij = [xj− 1

2, xj+ 1

2], where xj± 1

2are

called cell interfaces. The centre of each cell Ij is denoted by xj = 12(xj+ 1

2+xj− 1

2) and the value of

the function at the node xj is denoted by fj = f(xj). We assume that the set {xj+ 12}j is uniformly

spatial gridded throughout the domain and the cell size of Ij is denoted by 4x = xj+ 12− xj− 1

2.

The approximation of one-dimensional hyperbolic conservation laws (1) leads to system of ordinarydifferential equations by applying the method of lines, where the finite difference approximation isreplaced to the spatial derivative and yields a semi discrete scheme

dujdt

= − 1

4x(f̂j+ 1

2− f̂j− 1

2). (2)

Here uj is the approximation to the point value u(xj, t) and f̂j± 12

are called numerical fluxeswhich are Lipschitz continuous in each of its arguments and is consistent with the physical fluxf̂(u, ...., u) = f(u). The conservation property is obtained by defining a function h(x) implicitlythrough the following equation (see Lemma 2.1 of [25])

f(u(x, .)) =1

4x

xj+1

2∫xj− 1

2

h(ξ)dξ. (3)

Differentiating (3) with respect to x yields

f(u(x, .))x =1

∆x(h(xj+ 1

2)− h(xj− 1

2)), (4)

where h(xj± 12) is a approximation to the numerical flux f̂j± 1

2with a high order of accuracy, that

is,f̂j± 1

2= h(xj± 1

2) +O(∆xr). (5)

To ensure the numerical stability and to avoid entropy violating solutions, the flux f(u) is splittedinto two parts f+ and f−, thus

f(u) = f+(u) + f−(u), (6)

where df+(u)du≥ 0 and df−(u)

du≤ 0.

The numerical fluxes f̂+j+ 1

2

and f̂−j+ 1

2

evaluates at xj+ 12

obtained from (3) which are positive

and negative parts of f(u) respectively and with this we have f̂j+ 12

= f̂+j+ 1

2

+ f̂−j+ 1

2

. We will only

describe how f̂+j+ 1

2

is approximated because the negative part of the split flux, is symmetric to the

positive part with respect to xj+ 12. For brevity, we drop the ’+’ sign in the superscript from here

onwards.

4

2.1 Fifth order WENO schemes

The WENO schemes are designed for the approximation to the spatial derivative to solvethe hyperbolic conservation laws, which are used to reconstruct the unknown values of a givenflux function f from its known values in an essentially non-oscillatory manner. To construct f̂j+ 1

2,

the classical fifth-order WENO scheme uses five point stencil S5 = {xj−2, xj−1, xj, xj+1, xj+2}which is subdivided into three candidate sub-stencils S0(j),S1(j) and S2(j). To each cell Ij, thecorresponding stencil is denoted by

Sk(j) = {xj+k−2, xj+k−1, xj+k}, k = 0, 1, 2

and let

f̂kj+ 1

2=

2∑q=0

ck,qfj+k+q−2,

be the second-degree polynomial constructed on the stencil Sk(j) to approximate the value h(xj+ 12)

where the coefficients ck,q(q = 0, 1, 2) are the Lagrange’s interpolation coefficients depending onthe shifting parameter k. The flux values on each stencil can be written in the form of

f̂ 0j+ 1

2=

1

6(2fj−2 − 7fj−1 + 11fj),

f̂ 1j+ 1

2=

1

6(−fj−1 + 5fj + 2fj+1), (7)

f̂ 2j+ 1

2=

1

6(2fj + 5fj+1 − fj+2).

The flux f̂kj− 1

2

is obtained through shifting the index by −1. The Taylor expansion of (7) reveals

f̂ 0j± 1

2= hj± 1

2− 1

4f′′′

(0)4x3 +O(4x4),

f̂ 1j± 1

2= hj± 1

2+

1

12f′′′

(0)4x3 +O(4x4),

f̂ 2j± 1

2= hj± 1

2− 1

12f′′′

(0)4x3 +O(4x4).

The convex combination of these flux functions define the approximation to the value of h(xj+ 12)

which is

f̂j+ 12

=2∑

k=0

ωkf̂kj+ 1

2, (8)

where ωk are the non-linear weights. If the function h(x) is smooth in all the sub-stencils Sk(j),k = 0, 1, 2, we calculate the constants dk such that its linear combination with f̂k

j+ 12

gives the fifth

order convergence to h(xj+ 12), that is,

hj+ 12

=2∑

k=0

dkf̂kj+ 1

2+O(4x5).

The coefficients dk are known as the ideal weights because they generate the upstream centralfifth-order scheme for the five-point stencil. The values of ideal weights are given by

d0 = 1/10, d1 = 6/10, d2 = 3/10. (9)

5

The non-linear weights ωk are defined in (8) as

ωk =αk∑2q=0 αq

, (10)

and

αk =dk

(ε+ βk)2, (11)

where 0 < ε << 1 is introduced to prevent the denominator becoming zero and βk is a smoothnessindicator of the flux f̂k which measures the smoothness of a solution over a particular stencil. Thenecessary and sufficient conditions to achieve the fifth-order convergence for the WENO schemeare

2∑k=0

(ω±k − dk) = O(4x6), (12)

2∑k=0

Ak(ω+k − ω

−k ) = O(4x3), (13)

ω±k − dk = O(4x2). (14)

Since (12) holds always due to the normalization, a sufficient condition for the fifth-order conver-gence to the scheme is derived in [3] as

ω±k − dk = O(4x3), (15)

where the superscripts ′+′ and ′−′ on ωk correspond to their use in either fkj+ 1

2

and fkj− 1

2

respectively.

2.1.1 The WENO-JS scheme

The suggested smoothness indicators βk of Jiang and Shu in [17] are given by

βk =2∑

q=1

4x2q−1

x+4x2∫

x−4x2

(dqf̂k

dxq

)2

dx. (16)

The explicit form of the these smoothness indicators are as follows:

β0 =13

12(fj−2 − 2fj−1 + fj)

2 +1

4(fj−2 − 4fj−1 + 3fj)

2,

β1 =13

12(fj−1 − 2fj + fj+1)2 +

1

4(fj+1 − fj−1)2, (17)

β2 =13

12(fj − 2fj+1 + fj+2)2 +

1

4(3fj − 4fj+1 + fj+2)2.

By the Taylor’s expansion of these smoothness indicators, one obtain

β0 = f′24x2 + (

13

12f′′2 − 2

3f′f′′′

)4x4 + (−13

6f′′f′′′

+1

2f′f (4))4x5 +O(4x6),

β1 = f′24x2 + (

13

12f′′2

+1

3f′f′′′

)4x4 +O(4x6),

β2 = f′24x2 + (

13

12f′′2 − 2

3f′f′′′

)4x4 + (13

6f′′f′′′ − 1

2f′f (4))4x5 +O(4x6).

6

The smoothness indicators (17) are smoother than the smoothness indicators presented in [19],which are constructed based on the total variation measurement of a stencil in L1−norm. TheTaylor expansion of these smoothness indicators βk satisfies the sufficient condition

βk = D(1 +O(∆x2)),

where the constant D is independent of k but depends on the ∆x, which implies that the WENOweights satisfy the condition

ωk − dk = O(4x2).

Therefore, the final WENO reconstruction provides the fifth order convergence in the smoothregions. However at the critical points where the first derivative of f vanishes, the convergenceproperty may not hold since βk = D(1 + O(∆x)) yields ωk − dk = O(4x). Further, the order ofconvergence is degraded to two if the first derivative and second derivatives vanish but the thirdderivative is non-zero .

2.1.2 WENO-M

When the fifth-order WENO-JS scheme is used, the condition (15) may not hold at cer-tain smooth extrema or near critical points, yielding the third-order accuracy. To overcome thissituation Henrick et al. [15] introduced a mapping function gk(ω) as

gk(ω) =ω(dk + d2

k − 3dkω + ω2)

d2k + ω(1− 2dk)

, k = 0, 1, 2, (18)

where dk’s are the ideal weights and ω ∈ [0, 1]. This function is a non-decreasing monotone functionon [0, 1] which satisfies the following properties

(i) 0 ≤ gk(ω) ≤ 1, gk(0) = 0 and gk(1) = 1.

(ii) gk(ω) ≈ 0 if ω ≈ 0, gk(ω) ≈ 1 if ω ≈ 1.

(iii) gk(dk) = dk, g′k(dk) = g′′k(dk) = 0.

(iv) gk(ω) = dk +O(∆x6), if ω = dk +O(∆x2)

(19)

and with this mapping function, non-linear weights are defined as

ωMk =

αMk∑2

l=0 αMl

and αMk = gk(ωJS

k ), k = 0, 1, 2, (20)

where ωJSk are computed in (10) with the WENO-JS scheme by using the smoothness indicators

defined in (16).

2.1.3 WENO-Z

Borges et al. [3] presented a scheme, known as WENO-Z, by introducing global smoothnessindicator. The idea is to use whole five point stencil S5 to define a new smoothness indicator ofhigher order than the classical smoothness indicator βk, as

τ5 = |β0 − β2|. (21)

7

By Taylor’s expansion, the truncation error of τ5 is

τ5 =13

3|f ′′f ′′′|∆x5 +O(∆x6), (22)

and the smoothness indicators are defined as

βzk =

βk + ε

βk + ε+ τ5

, k = 0, 1, 2, (23)

and finally with this local smoothness indicators, the WENO weights are defined as

ωzk =

αzk∑2

q=0 αzq

, αzk =

dkβzk

= dk

(1 +

[τ5

βk + ε

]p), k = 0, 1, 2, (24)

where ε is chosen as a very small value in order to force this parameter to play only its originalrole of not allowing vanishing denominator for each weight and with the value p = 1 in (24), thescheme achieves the fourth order accuracy and with p = 2, the scheme achieves the fifth-orderaccuracy.

2.1.4 The WENO-NS scheme

Ha et al., have introduced smoothness indicators based on L1− norm in [9] which measuresthe smoothness of a solution on each 3-point stencil Sk(j), k = 0, 1, 2 by estimating the approximatemagnitude of derivatives as

βk = ξ|L1,kf |+ |L2,kf |, (25)

where the operators Ln,kf, k = 0, 1, 2 are the generalized undivided differences, defined by

|L1,kf | = (1− k)fj−2+k + (2k − 3)fj−1+k + (2− k)fj+k, (26)

|L2,kf | = fj−2+k − 2fj−1+k + fj+k. (27)

The number ξ is a parameter which is to balance the tradeoff between the accuracy aroundthe smooth regions and the discontinuous regions. The second term L2,kf is the same as the onesof the WENO-JS scheme, however, this scheme uses the absolute values while the later uses thesquared ones. The advantage with these operators Ln,kf is that the approximation of the derivative∆xnf (n) at the point, xj+ 1

2with the high order of accuracy can be obtained, that is,

Ln,kf =dnf

dxn(xj+ 1

2) +O(∆x3). (28)

By using Theorem 3.2 of [9], the Taylor’s expansion of β′ks are

β0 = ξ | ∆xf (1)

j+ 12

− 23

24∆x3f

(3)

j+ 12

| + | ∆x2f(2)

j+ 12

− 3

2∆x3f

(3)

j+ 12

| +O(∆x4),

β1 = ξ | ∆xf (1)

j+ 12

+1

24∆x3f

(3)

j+ 12

| + | ∆x2f(2)

j+ 12

− 1

2∆x3f

(3)

j+ 12

| +O(∆x4), (29)

β2 = ξ | ∆xf (1)

j+ 12

+1

24∆x3f

(3)

j+ 12

| + | ∆x2f(2)

j+ 12

+1

2∆x3f

(3)

j+ 12

| +O(∆x4).

8

The non-linear weights are defined as

ωNSk =

αNSk∑2

q=0 αNSq

, αNSk = dk

(1 +

ζ

(βk + ε)2

), k = 0, 1, 2, (30)

where

ζ =1

2(|β0 − β2|2 + g(L1,1f)2), g(x) =

x3

1 + x3(31)

so that the sufficient condition ω±k −dk = O(4x3) holds even first derivative vanish but the secondderivative is non-zero.

2.1.5 The WENO-P scheme

The drawback of WENO-NS scheme is that all the three sub-stencils S0(j),S1(j) and S2(j)may provide unbalanced contribution to the evaluation point xj+ 1

2because, the stencils S1(j)

and S2(j) are symmetric with respect to the point xj+ 12

whereas S0(j) is not. Therefore theinterpolation process may be over-influenced to the evaluation point from the left sub-stencils. Forthis, in [18] the authors improved the WENO-NS scheme by adjusting smoothness indicators tothe three sub-stencils by

β̃0 = β0, β̃1 = (1 + δ)β1, β̃2 = (1− δ)β2. (32)

They also observed that the global smoothness measurement which uses an additional contri-bution term which measures the regularity of a solution over the stencil S1(j). Instead of usingthe measurement proposed in WENO-NS scheme and to save computational cost, they proposedWENO-P scheme with modification to the weights as

ωPk =

αPk∑2

q=0 αPq

, αPk = dk

(1 +

ζ

(β̃k + ε)2

), k = 0, 1, 2, (33)

whereζ = (β0 − β2)2,

and these weights satisfy the sufficient condition to get fifth order accuracy when the first derivativevanishes but not the second derivative.

3 A new WENO scheme

To achieve the desired fifth-order accuracy when the first and second derivatives vanish butthe third derivative is non-zero, a modified WENO-P scheme called MWENO-P is proposed here.

9

The Taylor’s expansion of the operators Ln,kf defined in (26, 27) by using Theorem 3.2 of [9] are

L1,0f = ∆xf(1)

j+ 12

− 23

24∆x3f

(3)

j+ 12

+ ∆x4f(4)

j+ 12

+O(∆x5),

L1,1f = ∆xf(1)

j+ 12

+1

24∆x3f

(3)

j+ 12

+O(∆x5),

L1,2f = ∆xf(1)

j+ 12

+1

24∆x3f

(3)

j+ 12

+O(∆x5),

L2,0f = ∆x2f(2)

j+ 12

− 3

2∆x3f

(3)

j+ 12

+29

384∆x4f

(4)

j+ 12

+O(∆x5), (34)

L2,1f = ∆x2f(2)

j+ 12

− 1

2∆x3f

(3)

j+ 12

+5

24∆x4f

(4)

j+ 12

+O(∆x5),

L2,2f = ∆x2f(2)

j+ 12

+1

2∆x3f

(3)

j+ 12

+5

24∆x4f

(4)

j+ 12

+O(∆x5).

The main idea is here to construct a high-order global smoothness indicator which is to satisfythe sufficient condition (15) if the first and second derivatives vanish but the third derivative isnon-zero. Also the use of higher global smoothness indicator incurs the less dissipation near thediscontinuities in the numerical scheme [5]. To construct such a global smooth measurement, wedefine a variable η as

η = |L2,0f + L2,2f − 2L2,1f |2.

which is a linear combination of undivided differences of second-order derivatives leads to givea fourth-order of accuracy, so this leads to give a much smooth measurement than the globalsmoothness indicators presented in [9, 18]. The Taylor’s expansion of η yields

η = | − 51

384∆x4f

(4)

j+ 12

+O(∆x5)|2,

= ∆x8(A+O(∆x)2),

and we define the new non-linear weights as

ωMPk =

αMPk∑2

q=0 αMPq

, αMPk = dk

(1 +

η

(β̃k + ε)2

), k = 0, 1, 2. (35)

3.1 Convergence order at critical points

Here we discuss the convergence analysis of the proposed scheme MWENO-P at the criticalpoints, that is how the new weights ωMP

k are approaching the ideal weights dk in the presence ofcritical points. First, consider that there is no critical point and assume that ε = 0, then the localsmoothness indicators β̃k, as defined in (32) with weights as in (35) are of the form

β̃k =| ∆xf (1)

j+ 12

| + | ∆x2f(2)

j+ 12

| +O(∆x3) (36)

and the global smoothness indicator is of the form

η = O(∆x8) (37)

By substituting equations (36) and (37) in (35), the sufficient condition (15) immediately holds sothat the scheme has fifth convergence order.

10

If f′

j+ 12

= 0 and f′′

j+ 12

6= 0, the smoothness indicators are of the form

β̃k = |f ′′j+ 1

2|∆x2(1 +O(∆x))

then there is a constant D such that

1 +η

β̃k2 = 1 +D∆x4 +O(∆x5)

= (1 +D∆x4)

(1 +

O(∆x5)

1 +D∆x4

)= D∆x(1 +O(∆x5)) (38)

where D∆x = (1 + D∆x4) > 0. By substituting (38) in (35), then the sufficient condition holdshence the scheme has the fifth-order convergence at the first-order critical points.

Finally, if f′

j+ 12

= 0, f′′

j+ 12

= 0 and f′′′

j+ 12

6= 0, the smoothness indicators takes the form

β̃k = |f ′′′j+ 1

2|∆x3(1 +O(∆x)),

then there exist a constant D̂ such that

1 +η

β̃k2 = 1 + D̂∆x2 +O(∆x3)

= (1 + D̂∆x2)

(1 +

O(∆x3)

1 + D̂∆x2

)= D̂∆x(1 +O(∆x3)) (39)

where D̂∆x = (1+D̂∆x2) > 0. By substituting equation (39) in (35), one can see that the sufficientcondition holds and so the newly defined smoothness indicators attain the fifth-order convergenceeven in the case where first and second derivatives vanish but the third derivatives are non-zero.

4 Numerical Results

For time evaluation in (2) third-order TVD Runge-Kutta scheme (TVD RK3),

u(1) = un +4t L(un),

u(2) =3

4un +

1

4u(1) +

1

44t L(u(1)),

un+1 =1

3un +

2

3u(2) +

2

34t L(u(2)).

(40)

and fourth-order non-TVD Runge-Kutta schemes (RK-4),

u(1) = un +1

24t L(un),

u(2) = un +1

24t L(u1),

u(3) = un +4t L(u2),

un+1 =1

3(−un + u(1) + 2u(2) + u(3)) +

1

64t L(u(3)).

(41)

11

where L is the spatial operator, are used in following numerical examples for evaluating the ap-proximate solution using the proposed scheme MWENO-P.

The elaborate details of the schemes (40,41) can be found in [8]. The computed solution ofMWENO-P is compared with WENO-JS, WENO-Z, WENO-NS and WENO-P schemes. All thenumerical results are obtained on machine having an Intel(R) core (TM) i7 − 4700MQ processorwith 8 GB of memory. In order to ensure fairness in the comparison, all the schemes shared thesame subroutine calls and were compiled with the same compilation options. The only differencesbetween the implementation of the WENO schemes were on the subroutines for computing thecorresponding weights. In order to compare with the classical scheme, WENO-JS, we took ε =10−6 and ε = 10−40 for WENO-Z, WENO-NS, WENO-P and MWENO-P schemes in all thefollowing test cases. To prove the effectiveness of the scheme we considered various examples. Forconvergence analysis we first evaluated examples of scalar equation and later we have presentedresults for Euler equations.

4.1 Scalar test examples

The convergence analysis of the schemes is presented here by considering linear advectionand Burger’s equations with various initial profiles. Some of these initial profiles contain jumpdiscontinuity and in some cases, the solution in time leads to contact discontinuity and shock. Herewe’ve used the fifth-order flux version of WENO scheme based on Lax-Friedrich’s flux splittingtechnique and time step is taken as 4t ∼ (4x)

54 so that the fourth order non-TVD Runge-Kutta

method in time is effectively fifth-order. For all the examples in this subsection we’ve chosenξ = 0.1 to evaluate the equation (25) and δ = 0.05 to evaluate the equation (32).

4.1.1 Example 1:

Consider the transport equation

ut + ux = 0, −1 ≤ x ≤ 1, t ≥ 0, (42)

with the initial conditionu(x, 0) = sin(πx), (43)

and

u(x, 0) = sin(πx− 1

πsin(πx)), (44)

to test the numerical convergence of the proposed scheme. Below in Table 1 and Table 2 , the L1

and L∞ errors are calculated up to time t = 2 for WENO-JS, WENO-NS and WENO-P schemesalong with the proposed MWENO-P scheme for the initial condition (43). We observed thatMWENO-P scheme is converges slowly to the desired order of convergence for the approximatesolution. The initial condition (44) is a special case to test the order of convergence since its firstderivative vanish but second derivative is non-zero. Here too, the numerical order of convergencefor the MWENO-P scheme is more accurate when compared with WENO-JS, WENO-NS andWENO-P schemes which are shown in Table 3 and Table 4.

12

N WENO JS WENO-NS WENO-P MWENO-P10 4.8506e-02(—) 4.4448e-02(—) 5.4913e-02(—) 9.2794e-03(—)20 2.5414e-03(4.25) 2.1541e-03(4.36) 3.1845e-03(4.10) 3.0628e-04(4.92)40 8.9204e-05(4.83) 3.0544e-05(6.14) 5.0576e-05(5.97) 1.0249e-05(4.90)80 2.7766e-06(5.00) 3.1879e-07(6.58) 8.3516e-07(5.92) 3.1929e-07(5.00)160 8.6040e-08(5.01) 9.9380e-09(5.00) 1.3716e-08(5.92) 9.9414e-09(5.00)320 2.5528e-09(5.07) 3.1007e-10(5.00) 3.0641e-10(5.48) 3.1008e-10(5.00)

Table 1: L∞ errors of linear advection Eq. (42) with initial condition (43)

N WENO-JS WENO-NS WENO-P MWENO-P10 3.1593e-02(—) 3.1974e-02(—) 3.7323e-02(—) 5.2290e-03(—)20 1.5177e-03(4.37) 8.6517e-04(5.27) 1.3817e-03(4.75) 2.1156e-04(4.62)40 4.5188e-05(5.06) 9.8706e-06(6.45) 1.8070e-05(6.25) 6.6182e-06(4.99)80 1.4000e-06(5.01) 2.1181e-07(5.54) 2.0924e-07(6.43) 2.0420e-07(5.01)160 4.3621e-08(5.00) 6.3443e-09(5.06) 5.5174e-09(5.24) 6.3409e-09(5.00)320 1.3600e-09(5.00) 1.9759e-10(5.00) 1.8842e-10(4.87) 1.9757e-10(4.99)

Table 2: L1 errors of linear advection Eq. (42) with initial condition (43)





4.1.2 Example 2:

For the partial differential equation (42), consider the initial condition

u(x, 0) = sin(πx)3. (45)

13

For this initial condition, the first derivative and second derivatives vanish whereas the thirdderivative is non-zero. The L1 and L∞ errors along with the order of convergence for the MWENO-P scheme is shown in Table 5 and Table 6 respectively. The results shows the MWENO-P schemeachieves the desired order of accuracy where as WENO-NS and WENO-P fails to achieve.

N WENO-JS WENO-NS WENO-P MWENO-P10 1.8058e-01(—) 1.5556e-01(—) 1.4980e-01(—) 1.6087e-01(—)20 6.3274e-02(1.51) 3.1225e-02(2.31) 3.3257e-02(2.17) 2.6674e-02(2.59)40 6.0354e-03(3.39) 6.6848e-03(2.22) 8.1907e-03(2.02) 3.0858e-03(3.11)80 9.1031e-04(2.72) 9.1944e-04(2.86) 1.2103e-03(2.75) 3.0617e-04(3.33)160 4.8182e-05(4.23) 1.0142e-04(3.18) 1.3156e-04(3.20) 1.3278e-06(7.84)320 8.0849e-07(4.89) 9.8843e-06(3.35) 1.3798e-05(3.25) 3.8780e-08(5.09)640 1.3257e-08(5.93) 9.9097e-07(3.31) 1.3735e-06(3.32) 1.1322e-09(5.09)1280 2.3166e-10(5.83) 9.9188e-08(3.32) 1.4340e-07(3.26) 3.4832e-11(5.02)


N WENO-JS WENO-NS WENO-P MWENO-P10 2.4666e-01(—) 2.2154e-01(—) 2.1397e-01(—) 2.2637e-01(—)20 1.2969e-01(0.92) 6.0015e-02(1.88) 7.2114e-02(1.56) 4.7482e-02(2.25)40 1.2970e-02(3.32) 1.6773e-02(1.83) 2.1148e-02(1.77) 1.1456e-02(2.05)80 3.8067e-03(1.76) 3.7616e-03(2.15) 5.3389e-03(1.98) 2.2518e-03(2.34)160 3.4089e-04(3.48) 8.0170e-04(2.23) 1.0032e-03(2.41) 4.2414e-06(9.05)320 6.9825e-06(5.60) 1.4264e-04(2.49) 1.9960e-04(2.32) 9.6777e-08(5.45)640 7.5668e-08(6.52) 2.4583e-05(2.53) 3.6812e-05(2.43) 1.7514e-09(5.78)1280 7.2140e-10(6.71) 4.7171e-06(2.38) 6.7828e-06(2.44) 5.4793e-11(4.99)


4.1.3 Example 3:

For linear advection equation (42), let the initial condition be

u(x, 0) = u0(x) =

{−sin(πx)− 1

2x3 for − 1 ≤ x < 0,

−sin(πx)− 12x3 + 1 for 0 ≤ x < 1.

(46)

which is a piecewise sine function with jump discontinuity at x = 0. The solution is computed withthe CFL number 0.5 with uniform concretization of the domain and the step size is 4x = 0.01up to time t = 8. The approximate solution computed with MWENO-P along with WENO-JS,WENO-NS and WENO-P schemes is plotted in Figure 1 against the exact solution. It can beobserved from the plot that the proposed scheme performs better than other schemes near thejump discontinuity.

14

Figure 1: Numerical solution of linear advection Eq. (42) with the initial condition (46)

4.1.4 Example 4:

Consider the discontinuous profile

u(x, 0) =

{1 if − 0.5 ≤ x < 0.5,

0 otherwise.(47)

for the linear equation (42). The equation is solved with the CFL number 0.5 with the spatial stepsize 4x = 0.01. For time t = 10 the computed approximate solutions are plotted against exactsolution in Figure 2. The proposed scheme MWENO-P has better approximation than WENO-JS,WENO-NS and WENO-P schemes especially near the areas of discontinuities.

Figure 2: Numerical solution of linear advection Eq. (42) with the initial condition (47)

15

4.1.5 Example 5:

The WENO schemes are designed as a shock capturing schemes for solving the hyperbolic conser-vation laws, for this as a last example in this section considered the Burger’s equation

ut + (u2

2)x = 0, −1 ≤ x ≤ 1, t ≥ 0, (48)

u(x, 0) = u0(x).

subject to periodic boundary conditions. In Figure 3, the numerical result of the fifth-order WENOschemes for the initial conditions

u0(x) = −sin(πx). (49)

at time t = 1.5 and

u0(x) =1

2+ sin(πx). (50)

at t = 0.55 are plotted respectively. The exact or reference solution is calculated with 2000 gridpoints with WENO-JS scheme and the approximate solutions are computed with 200 grid pointsin space. It is shown that the shocks are very well captured by all the schemes.

Figure 3: Approximate solution of (48) with initial condition (49) and (50)

4.2 Euler equations in one space dimension

The one-dimensional Euler equations are given by ρρuE

t

+

ρuρu2 + pu(E + p)

x

= 0 (51)

16

where ρ, u, E, p are the density, velocity, total energy and pressure respectively. The system (51)represents the conservation of mass, momentum and energy. The total energy for an ideal poly-tropic gas is defined as

E =p

γ − 1+

1

2ρu2,

where γ is the ratio of specific heats and its value is taken as γ = 1.4.The numerical flux f̂j+ 1

2at xj+ 1

2is calculated based on the steps given in [22], which are

reproduced here for completeness.

• Compute an average state uj+ 12

using Roe’s mean matrix.

• Compute the left eigenvectors, the right eigenvectors and the eigenvalues of the Jacobianf′(u) at the average state uj+ 1

2and denote them by L = L(uj+ 1

2), R = R(uj+ 1

2), and

∧ = ∧(uj+ 12) respectively. Note that L = R−1.

• Projecting the flux and the solution which are in the potential stencil of the WENO re-construction for obtaining the flux f̂j+ 1

2to the local characteristic fields by sj = R−1uj,

qj = R−1f(uj), j in a neighborhood of i.

• To obtain the corresponding component of the flux q̂±j+ 1

2

, Lax-Friedrich’s flux splitting and the

WENO reconstruction procedure is used for each component of the characteristic variables.

• Projecting back the characteristic variables to physical variables by f̂±j+ 1

2

= Rq̂±j+ 1

2

.

• Finally, form the flux by taking f̂j+ 12

= f̂+j+ 1

2

+ f̂−j+ 1

2

.

For time evaluation, we used third-order TVD Runge-Kutta scheme (40). Now consider the onedimensional Riemann problem for Euler system of equations (51) i.e., with the initial condition

U(x, 0) =

{UL if x < x0,

UR if x ≥ x0.

where UL = (ρl, ul, pl) and UR = (ρr, ur, pr). In the following, various test cases are taken fornumerical study of MWENO-P scheme.

4.2.1 Sod’s shock tube problem:

For this the initial condition is given by

U(x, 0) =

{(1.0, 0.75, 1.0), if 0 ≤ x ≤ 0.5,

(0.125, 0.0, 0.1), if 0.5 ≤ x ≤ 1.

This is an example of modified version of Sod’s problem defined in [26] and its solution containsa right shock wave, a right traveling contact wave and a left sonic rarefaction wave. Transmissiveboundary conditions are taken for numerical evaluation. The solution is computed up to timet = 0.2 with 200 grid points in space with the CFL number 0.5. The density and pressure profilefor various fifth-order WENO schemes are shown in Figure 4 and Figure 5 respectively againstthe reference solution which is calculated with 2000 grid points using WENO-JS scheme. It isobserved that the proposed scheme MWENO-P performs better than other WENO schemes nearthe region of contact discontinuity.

17

Figure 4: Density profile for Sod’s problem and the zoomed region at the contact discontinuity

Figure 5: Pressure profile of Sod’s shock tube problem

4.2.2 Shock tube problem with Lax initial condition:

For this the initial condition which is given by

U(x, 0) =

{(0.445, 0.698, 3.528), if − 5 ≤ x < 0,

(0.500, 0.000, 0.571), if 0 ≤ x ≤ 5.

For this, the solution is computed up to time t = 1.3 with 200 grid points along space directionand the CFL number is set to be 0.5. The reference solution is calculated with 2000 grid pointsby using WENO-JS scheme. For assessing the performance of MWENO-P scheme, the densityprofile is plotted against the reference solution along with other schemes in Figure 6, it can be

18

seen that MWENO-P scheme performs better than other schemes near the discontinuities. Thepressure profile is displayed in Figure 7.

Figure 6: Density profile for Lax initial condition

Figure 7: Pressure profile for Lax initial condition

4.2.3 1D shock entropy wave interaction problem:

The initial condition as given in [27] for shock entropy wave interaction problem is

U(x, 0) =

{(3.857143, 2.629369, 10.33333), if − 5 ≤ x < −4,

(1 + εsin(kx), 0.000, 1.000), if − 4 ≤ x ≤ 5.

19

where ε and k are the amplitude and wavenumber of the entropy wave respectively, chosen ε = 0.2and k = 5. This problem has a right-moving supersonic (Mach 3) shock wave which interacts withsine waves in a density disturbance, that generates a flow field with both smooth structures anddiscontinuities. This flow induces wave trails behind a right-going shock with wave numbers higherthan the initial density-variation wavenumber k. Since the exact solution is unknown, the referencesolution with zero gradient boundary conditions is obtained by using the fifth-order WENO-JSscheme with 2000 grid points. The initial condition contains a jump discontinuity at x = −4 andespecially, the initial density profile has oscillations on [−4, 5]. The solution is computed up to timet = 1.8 with 200 spatial grid points and the CFL number is set to be 0.5. The Figure 8 containsthe graph of the solutions calculated at 200 grid points and its zoomed region near oscillations. Itis observed that MWENO-P scheme performs better than WENO-JS and WENO-P at 200 gridpoints.

Figure 8: Shock entropy wave interaction test with 200 grid points

4.3 Two-Dimensional Euler equations

In this section, we apply the proposed scheme to two-dimensional problem in cartesian coor-dinates. The governing two-dimensional compressible Euler equations is given by

Ut + F (U)x +G(U)y = 0

where U = (ρ, ρu, ρv, E)T , F (U) = (ρu, P +ρu2, ρuv, u(E+P ))T , G(U) = (ρv, ρuv, P +ρv2, v(E+P ))T . The total energy E and the pressure p is defined by

p = (γ − 1)(E − 1

2ρ(u2 + v2))

where γ is the ratio of specific heats. Here ρ, u, v are density, x-wise-velocity component andy-wise-velocity component respectively.

20

4.3.1 2D problem of 2D gas dynamics

The 2D Riemann problem of gas dynamics is proposed in [21] and solved on the rectangulardomain [0, 1]×[0, 1]. The 2D Riemann problem is defined by initial constant states in each quadrantwhich is divided by lines x = 0.8 and y = 0.8 on the square as

(ρ, u, v, p) =

(1.5, 0, 0, 1.5) if 0.8 ≤ x ≤ 1, 0.8 ≤ y ≤ 1,

(0.5323, 1.206, 0, 0.3) if 0 ≤ x < 0.8, 0.8 ≤ y ≤ 1,

(0.138, 1.206, 1.206, 0.029) if 0 ≤ x < 0.8, 0 ≤ y < 0.8,

(0.5323, 0, 1.206, 0.3) if 0.8 < x ≤ 1, 0 ≤ y ≤ 0.8,

with Dirichlet boundary conditions. According to the initial conditions, four shocks come intobeing and produce a narrow jet. The numerical solution is calculated with 400 × 400 grid pointsup to t = 0.8 with the CFL number 0.5. An examination of these results reveals in figure 9that, MWENO-P yields a better solution of the complex structure appearing when compares toWENO-JS and WENO-P schemes.

Figure 9: Density profile of 2D Riemann problem of 2D gas dynamics with the mesh ∆x = ∆y = 1/400

4.3.2 Two-Dimensional Rayleigh-Taylor instability

The Taylor instability happens on an interface between two fluids of different densities whenan acceleration is directed from heavier fluid to lighter fluid. This problem has been simulatedextensively in the literature [28]. The computational domain is [0, 1/4] × [0, 1] and the initialconditions are

(ρ, u, v, p) =

{(2, 0,−0.025acos(8πx), 2y + 1) if 0 ≤ y < 0.5,

(1, 0,−0.025acos(8πx), y + 32) if 0.5 ≤ y < 1.

with the sound speed a =√γp/ρ and the ratio of specific heats γ = 5/3. The gravitational effect

is introduced by adding ρ and ρv to the right hand side of third equation and fourth equation,

21

respectively of the two-dimensional Euler equation. Reflective boundary conditions are imposedfor the left and right boundaries and

(ρ, u, v, p) =

{(2, 0, 0, 2.5) top boundary,

(1, 0, 0, 1) bottom boundary.

The final simulation time is t = 1.95. The density contour plotted in Fig. 10 shows that theWENO-P and MWENO-P obtains more complex structures than the other schemes.

Figure 10: Density profiles of Rayleigh-Taylor instability problem at t=1.95 with the mesh ∆x = ∆y = 1/500

4.3.3 Double Mach reflection of a strong shock

The final test considered in this paper is the two-dimensional double mach reflection problem[27]where a vertical shock wave moves horizontally into a wedge that is inclined by some angle. Thecomputational domain for this problem is chosen to be [0, 4]× [0, 1], and the reflecting wall lies atthe bottom of the computational domain for 1

6≤ x ≤ 4. Initially a right-moving Mach 10 shock is

positioned at x = 16, y = 0, and makes a 60o angle with the x-axis. For the bottom boundary, the

exact postshock condition is imposed for the part from x = 0 to x = 16

and a reflective boundarycondition is used for the rest. The top boundary of our computational domain uses the exactmotion of the Mach 10 shock. Inflow and outflow boundary conditions are used for the left andright boundaries. The unshocked fluid has a density of 1.4 and a pressure of 1. The problem wasrun till t = 0.2 and the blow-up region around the double Mach stems. The ration of specific heatsγ = 1.4 and we set CFL number as 0.5. The results in [0, 3]× [0, 1] are displayed for the WENO-JS,WENO-P and MWENO-P schemes in figures 11, 12 and 13 respectively. The figure 14 shows thatthe performance of the schemes WENO-JS, WENO-P and MWENO-P at the Mach stem of thedensity variable at the final time with several grid points. It can be clearly seen that MWENO-Presolves better the instabilities around the Mach stem.

22

Figure 11: Density profiles of Double Mach reflection of a strong shock with WENO-JS scheme at t=0.2 with the mesh ∆x = ∆y = 1/400

Figure 12: Density profiles of Double Mach reflection of a strong shock with WENO-P scheme at t=0.2 with the mesh ∆x = ∆y = 1/400

Figure 13: Density profiles of Double Mach reflection of a strong shock with MWENO-P scheme at t=0.2 with the mesh ∆x = ∆y =1/400

23

Figure 14: Density profiles of Double Mach reflection of a strong shock at Mach stem with WENO-JS, WENO-P and MWENO-Pschemes at t=0.2 with the mesh ∆x = ∆y = 1/400

5 Conclusion

In this paper, a modified fifth-order WENO scheme to approximate the solution of nonlinearhyperbolic conservation laws named as MWENO-P was presented, by introducing a new globalsmoothness indicator based on undivided differences of second order derivatives of an interpola-tion polynomial over a stencil. The motivation to the study is that the WENO-NS and WENO-Pschemes do not satisfy the sufficient condition if first-order derivatives vanish but the secondderivative is non-zero. The proposed scheme satisfied the sufficient condition even if first andsecond derivatives vanish but the third derivative is non-zero. The approximate solutions to theone-dimensional scalar, system and two-dimensional system of hyperbolic conservation laws aresimulated with the proposed scheme and compared it with other fifth-order WENO schemes. Nu-merical experiments show that the proposed scheme yields better approximation than other fifth-order schemes, especially to the numerical problems which contain discontinuities while keepingessentially non-oscillatory performance.

Acknowledgements: The authors are gratefully thank to the National Board for Higher Math-ematics (NBHM) for the financial support by the Grant Reference No. 2/48(4)/2014/NBHM/R& D II//14356 through the Department of Atomic Energy (DAE), India.

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A modi ed fth-order WENO scheme for hyperbolic ... · PDF fileA modi ed fth-order WENO scheme for hyperbolic conservation laws Samala ... at the critical points where the rst derivatives